Fast diffusion-controlled coagulation

If we remove the electrostatic barrier, e.g. by ion adsorption or by adding electrolyte, we can obtain a fast coagulation that is controlled by the difiusion mechanism and where we can assume that all collisions lead to adhesion of molecules. Smoluchowski has derived the following equation for the rate constant in the case of fast coagulation  

In this case the second-order constant does not depend on particle size, only on temperature and the viscosity of medium. This rate constant often implies rapid colloid aggregation. 

Eqpiation 11.3 is approximate but gives an idea of how fast the coagulation can be. An important measure of the speed of coagulation is the so-called half-life time (see Problem 11.5), the time when the number of particles is decreased to half  

For concentrated dispersions, the half-life time is in the milliseconds range, while for more usual colloidal systems typical values are higher, of the order of seconds m minutes. 

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In the absence of a barrier to coagulation, and if the primary minimum is deep, every collision between a particle and a floe will lead to the growth of the floe. The rate of coagulation is then controlled entirely by the kinetics of the diffusion process leading to particle-particle collision. The theory of fast coagulation was developed originally by Smoluchowski (1918) and elaborated by Muller (1926). The rate equation has the same form as that for a bimolecular reaction ... 

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